Doolittle Equation

Why do materials like honey or polymers become immensely thick and resistant to flow as they cool, a phenomenon far more dramatic than in simple liquids like water? This question points to a central puzzle in materials science and physics. The answer lies not in complex molecular forces, but in a surprisingly simple and elegant concept: free volume. The Doolittle equation provides the mathematical foundation for this idea, linking a material's viscosity directly to the amount of empty space available for its molecules to move into. This article delves into this powerful model, addressing the gap between long-standing empirical observations and a cohesive physical explanation for the behavior of glass-forming liquids.

Across the following chapters, you will gain a comprehensive understanding of this cornerstone theory. The first section, ​​Principles and Mechanisms​​, will build the concept from an intuitive analogy to its quantitative formulation, showing how it explains the glass transition and unifies the famous Doolittle, VFT, and WLF equations into a single, coherent story. Following this, the chapter on ​​Applications and Interdisciplinary Connections​​ will showcase the theory's remarkable utility, from designing plastics with specific properties to understanding phenomena in fields as varied as metallurgy and electrochemistry, demonstrating how a simple physical picture can unlock complex problems across science and engineering.

Why does honey get thicker when you put it in the fridge? This question, in a more general sense, is one of the deepest puzzles in the physics of matter. For everyday liquids like water, the change in "stickiness"—or ​​viscosity​​, as physicists call it—is noticeable but not extreme. But for other materials, like the polymers that make up our plastics or the silica that forms glass, the change is staggering. As they cool, their viscosity can increase by a factor of a trillion or more over a relatively small temperature range, transforming them from a flowing liquid into a rigid, amorphous solid—a glass. What is the secret behind this dramatic slowdown? The answer lies in a beautifully simple yet powerful idea: the concept of ​​free volume​​.

Imagine you are in a crowded ballroom. If people are packed shoulder-to-shoulder, moving from one side of the room to the other is nearly impossible. You are stuck. Now, imagine a few people leave, creating some empty space. Suddenly, you can move by stepping into an adjacent empty spot, and the person behind you can step into the spot you just left. A chain reaction begins, and the crowd can start to flow. The more empty space, the easier it is for everyone to move.

This is the essence of the ​​free volume theory​​. The molecules in a liquid are like the dancers in the ballroom. For the liquid to flow, its molecules must be able to move past one another. This movement isn't a free-for-all; it happens through a series of discrete jumps. A molecule can only jump if there is a vacant spot, or a "hole," of a sufficient size right next to it. The collection of all these tiny holes constitutes the liquid's ​​free volume​​. The resistance to flow—the viscosity—is therefore determined not by the speed of the molecules themselves, but by the probability of finding a hole to jump into. When a liquid is hot, the molecules are energetic and jiggle around, creating plenty of transient gaps. As it cools, the system packs more densely, the free volume shrinks, and the dancers find themselves gridlocked.

So, how can we turn this intuitive picture into a mathematical law? This is where the insight of Arthur K. Doolittle comes in. He proposed a relationship that captures the core of the free volume idea. Let's call the intrinsic volume occupied by the hard cores of the molecules $V_0$ (the space taken by the dancers) and the total empty space $V_f$ (the free volume). Doolittle suggested that the viscosity, $\eta$, depends exponentially on the ratio of the occupied volume to the free volume. More precisely, he found that the natural logarithm of the viscosity is linearly related to this ratio:

Here, $A$ and $B$ are constants for a given material, with $B$ typically being a number close to one. This is the celebrated ​​Doolittle equation​​. At first glance, it might seem abstract, but its physical meaning is profound. The logarithm on the left side tells us that we should think about viscosity in multiplicative terms (factors of 10, 100, etc.), which is exactly what we observe experimentally. The ratio $V_0 / V_f$ on the right side is the heart of the matter. If the free volume $V_f$ becomes very small compared to the occupied volume $V_0$, this ratio becomes enormous. The exponential nature of the relationship means that even a small decrease in free volume can cause a gigantic, "catastrophic" increase in viscosity. This is precisely the mechanism needed to explain the dramatic thickening of glass-forming liquids. The equation is a direct mathematical translation of our ballroom analogy: as the empty space vanishes, the difficulty of moving skyrockets.

The Doolittle equation speaks of volumes, but our initial puzzle was about temperature. The link is straightforward: materials expand when heated and contract when cooled. So, the free volume itself must depend on temperature. Experiments show that for many amorphous materials above their ​​glass transition temperature​​, $T_g$, the ​​fractional free volume​​, $f = V_f / (V_0 + V_f)$, increases in a simple, linear fashion with temperature. We can write this as:

where $f_g$ is the very small fractional free volume that gets "frozen in" at the glass transition temperature, and $\alpha_f$ is the thermal expansion coefficient of the free volume—a measure of how quickly new free space is created as we raise the temperature.

Now we have a complete logical chain: changing the temperature changes the free volume, and according to the Doolittle equation, changing the free volume causes a dramatic change in viscosity. By combining these two equations, we can predict how sticky a polymer will be at a manufacturing temperature, given its properties at the glass transition temperature. This is not just an academic exercise; it's a critical tool for industries that shape and mold polymers.

Let's push this idea to its limit. If free volume increases linearly upon heating, it must decrease linearly upon cooling. What if we keep cooling the liquid, far below its usual freezing point, without letting it crystallize? Our linear model, in a slightly simplified form $f(T) = \alpha_f (T - T_0)$, suggests that the free volume would shrink and shrink until it completely vanishes at some finite, positive temperature, $T_0$.

What would happen at this ​​Vogel temperature​​, $T_0$? If $f(T)$ goes to zero, the Doolittle equation predicts that the viscosity $\eta$ must diverge to infinity! By substituting the linear model for $f(T)$ into the Doolittle equation, we derive another famous relationship, the ​​Vogel-Fulcher-Tammann (VFT) equation​​:

where $A'$ and $B'$ are new constants. This equation suggests a "kinetic catastrophe" at $T_0$, where all molecular motion would cease and the liquid would become infinitely rigid.

Does this divergence actually happen? No. Reality intervenes. As a liquid cools, its relaxation time (which is proportional to its viscosity) gets longer and longer. At some point, this relaxation time becomes longer than the time we are willing to wait in an experiment (say, a few minutes). The liquid can no longer rearrange its structure to stay in thermal equilibrium. It falls out of equilibrium and its structure becomes "frozen." This is the glass transition, and it happens at a temperature $T_g$ which is always higher than the idealized divergence temperature $T_0$. A small amount of residual free volume is trapped in the glass, preventing the viscosity from ever truly becoming infinite. The divergence at $T_0$ is thus a theoretical specter, a mathematical ghost that haunts the liquid, signaling the impending doom of kinetic arrest but never quite catching its prey.

For decades, engineers working with polymers used a powerful empirical tool called ​​time-temperature superposition​​. They found that the viscoelastic behavior of a polymer at a high temperature over a short time is equivalent to its behavior at a low temperature over a long time. The relationship between time and temperature could be captured by a ​​shift factor​​, $a_T$. They could measure the properties at one reference temperature (usually $T_g$) and then use a "master curve" and the shift factor to predict the properties at any other temperature.

This shift factor was famously described by the empirical ​​Williams-Landel-Ferry (WLF) equation​​:

where $C_1$ and $C_2$ were fitting constants. For a wide variety of polymers, these constants were found to have nearly "universal" values ($C_1 \approx 17.4$ and $C_2 \approx 51.6 \text{ K}$). Was this just a lucky coincidence?

Here lies the true beauty of a good physical model. It connects and unifies seemingly disparate observations. By defining the shift factor as the ratio of viscosities, $a_T = \eta(T)/\eta(T_g)$, and using our Doolittle free volume model, we can derive the WLF equation from first principles! The derivation shows that the complicated functional form of the WLF equation is a direct mathematical consequence of how free volume changes with temperature.

Furthermore, this derivation reveals the physical meaning of the "universal" constants. They are not arbitrary numbers, but combinations of the fundamental free volume parameters:

The constants are "universal" simply because many common polymers happen to share similar free volume properties near their glass transition, namely a fractional free volume $f_g$ of about $2.5\%$ and a similar expansion coefficient $\alpha_f$. The WLF equation, once a clever empirical trick, is revealed to be a deep consequence of the underlying physics of free volume, a testament to the unifying power of Doolittle's simple idea. This framework also allows us to predict how and why the WLF constants would deviate for polymers with different structures, like those with strong hydrogen bonds or extensive crosslinking.

Finally, the free volume theory helps us understand another key property of glasses: their ​​fragility​​. This term doesn't refer to mechanical breakability, but to how violently the viscosity changes upon cooling. "Strong" liquids, like window glass, show a gradual increase in viscosity over a wide temperature range. "Fragile" liquids, like many polymers, behave like normal liquids until they get close to $T_g$, at which point their viscosity shoots up almost vertically.

The Doolittle equation provides an elegant explanation. The sensitivity of viscosity to changes in free volume is immense, especially as $f$ becomes small. Our derivation of the fragility index, a quantitative measure of this sharpness, shows that it is inversely proportional to the square of the free volume at the glass transition, $m \propto 1/f_g^2$. Fragile liquids are those that operate with a very small margin of free volume near $T_g$. Any slight decrease in temperature, which causes a tiny reduction in the already-scarce free volume, has a catastrophic effect on mobility. They are living on the edge of a kinetic cliff.

This free volume picture, born from a simple analogy of a crowded room, provides a remarkably coherent framework. It explains the colossal change in viscosity near the glass transition, connects it to temperature, and unifies the Doolittle, VFT, and WLF equations into a single story. However, like all great theories, it is important to know its boundaries. The model exquisitely describes the cooperative, large-scale molecular rearrangements that dominate above $T_g$. Well below $T_g$, in the frozen glassy state, these motions are arrested. The physics changes. Any residual molecular wiggles are small, local events that follow a simpler, thermally-activated law known as the Arrhenius equation. The Doolittle/WLF world gives way to the Arrhenius world, reminding us that even the most beautiful theories have their domain of truth.

We have spent some time exploring the rather simple and intuitive idea that in the densely packed world of an amorphous material, the ability for anything to move—to flow, to relax, to diffuse—is dictated by the amount of empty space, or "free volume," that exists between the molecules. The Doolittle equation, $\eta = A \exp(B/f)$, is the beautifully concise mathematical statement of this idea. At first glance, it might seem like just another formula. But it is much more. It is a key that unlocks a surprisingly vast and varied landscape of phenomena, from the industrial production of plastics to the fundamental physics of glasses and the design of next-generation technologies. Now that we understand the principle, let's take a journey to see what it can do.

Long before the free volume theory was fully developed, scientists who studied polymers and other glass-forming liquids had noticed a dramatic, almost violent, change in viscosity with temperature. They came up with empirical formulas to describe what they saw. Two of the most famous are the Vogel-Fulcher-Tammann (VFT) equation and the Williams-Landel-Ferry (WLF) equation. For a long time, these were just exceptionally useful curve-fitting tools. But the Doolittle equation shows us they are much more; they are direct consequences of our free volume picture.

Let’s imagine the free volume fraction, $f$, increases linearly as we heat the material above some characteristic temperature, say the glass transition temperature $T_g$. A simple model for this is $f(T) = f_g + \alpha_f (T - T_g)$, where $f_g$ is the free volume at $T_g$ and $\alpha_f$ is its thermal expansion coefficient. If you substitute this expression for $f$ into the Doolittle equation, a little bit of algebraic rearrangement reveals something remarkable. The equation transforms into the VFT form, $\eta \propto \exp(B_{\text{VFT}}/(T - T_0))$, where the famous Vogel temperature, $T_0$, is no longer just a fitting parameter but has a physical meaning: it's the hypothetical temperature where the free volume would vanish entirely, bringing all motion to a grinding halt.

The WLF equation, a cornerstone of polymer engineering, is revealed to be the very same idea in a different set of clothes. It simply expresses the change in viscosity (or relaxation time) relative to a reference temperature, usually $T_g$. The Doolittle model allows us to derive the WLF equation from first principles and, in doing so, gives physical meaning to the WLF constants, $C_1$ and $C_2$. They are not arbitrary; they are determined by the material's fundamental free volume characteristics: $C_1$ is related to $B$ and the free volume at the glass transition, $f_g$, while $C_2$ is the ratio of $f_g$ to the free volume's expansion coefficient, $\alpha_f$. This also explains why the so-called "universal" WLF constants work for many polymers but fail for others. If a new polymer has an unusual free volume expansion coefficient, its WLF constants will be different, a fact we can now predict quantitatively. The free volume model takes us from empirical rules to physical understanding.

If you've ever wondered why a vinyl shower curtain is flexible while a PVC pipe is rigid, you've encountered the art of plasticization. A plasticizer is a small molecule that, when mixed into a polymer, makes it softer and more pliable. How does it work? In our framework, the answer is simple: the small plasticizer molecules act as "molecular spacers," efficiently creating extra free volume in the material. This increased free volume lowers the glass transition temperature and dramatically reduces the viscosity at any given temperature.

The Doolittle framework allows us to turn this qualitative picture into a predictive science. By assuming simple mixing rules for how the free volumes of the polymer and plasticizer combine, we can predict how the properties of the mixture, such as its WLF parameters, will change with the plasticizer concentration.

This predictive power is not just an academic exercise; it is the bread and butter of materials engineering. Imagine you need to design a polymer for injection molding. The material must be fluid enough to flow into the mold at a high processing temperature, but it must become a stable solid at its lower service temperature. This means its viscosity, $\eta$, must be below a certain threshold during processing, and its glass transition temperature, $T_{g,\mathrm{mix}}$, must remain above the operating temperature. By combining the Doolittle equation (for viscosity), a model for free volume versus temperature, and a model for how the plasticizer affects $T_{g,\mathrm{mix}}$ (like the Fox equation), an engineer can calculate the precise, optimal amount of plasticizer needed to satisfy both constraints simultaneously. This is the power of a good physical model: it turns guesswork into design.

What happens if we take our amorphous material and squeeze it? Intuitively, applying hydrostatic pressure will compress the material, and this compression will disproportionately squeeze out the "empty" free volume. According to the Doolittle equation, a decrease in $f$ should cause an exponential increase in viscosity. And this is exactly what happens.

Our free volume theory can make this quantitative. By defining a "compressibility of the free volume," we can derive an expression for the pressure coefficient of viscosity—a measure of how sensitive the material's flow is to pressure. For polymers, this sensitivity is enormous. A change in pressure that might be barely noticeable in the density can cause the viscosity to increase by many orders of magnitude.

Consider a polymer melt in an injection molding machine. At atmospheric pressure, it might be a viscous liquid with a characteristic relaxation time of less than a second. But when the pressure is ramped up to hundreds of atmospheres, the free volume is squeezed out. Our model predicts that this change can cause the relaxation time to leap to hours, days, or even years. The liquid, for all practical purposes, has been momentarily "frozen" into a solid simply by squeezing it. This phenomenon is of immense practical importance in polymer processing and also offers a window into the behavior of materials deep within the Earth's crust.

The true beauty of a fundamental scientific idea is its ability to reach across fields, connecting seemingly disparate phenomena. The Doolittle equation and the free volume concept are a perfect example of this unity.

• ​​Solid-State Physics & Metallurgy:​​ The problem of flow is not unique to polymers. Metallic glasses are alloys frozen into an amorphous state, much like a polymer glass. The atoms in these materials are also in a crowded jumble, and for one to move, it must find a pocket of free volume. We can combine the Doolittle equation for viscosity with the famous Stokes-Einstein relation, which links viscosity to diffusion. This allows us to derive an expression for the self-diffusion coefficient of an atom in a metallic glass, connecting a microscopic atomic property (diffusion) to a macroscopic material property (viscosity) through the single concept of free volume.

• ​​Polymer Dynamics:​​ For very long, entangled polymer chains, the dynamics are often described by reptation theory, which imagines a chain slithering like a snake through a "tube" formed by its neighbors. But what determines the friction the snake feels as it slithers? It is the friction experienced by its individual segments. This "monomeric friction" is itself governed by the local availability of free volume. Thus, the Doolittle equation provides a crucial sub-model within reptation theory, describing how the fundamental step of motion changes with temperature and pressure, thereby controlling the overall disengagement time of the entire chain.

• ​​Electrochemistry & Mechanics:​​ Perhaps one of the most exciting modern frontiers is in flexible electronics, such as bendable batteries. Many of these devices rely on polymer electrolytes, where ions move through a solid polymer matrix. This ion transport is a form of diffusion and, you guessed it, is governed by free volume. The Doolittle equation can be adapted to describe ionic conductivity. This leads to a stunning prediction: if you take a polymer electrolyte and stretch it, you increase its total volume. Assuming this extra volume becomes free volume, the conductivity should increase. The theory allows us to connect a mechanical property, the Poisson's ratio (which describes how the material deforms), to an electrical property, the change in conductivity, all via the underlying free volume. This is a remarkable intersection of mechanics, electrochemistry, and materials science, all unified by one simple idea.

From the viscosity of everyday plastics to the diffusion of atoms in exotic metals and the performance of futuristic batteries, the Doolittle equation shows its power. It reminds us that sometimes the most profound insights come from the simplest physical pictures—in this case, the realization that to move in a crowd, what you really need is a little bit of elbow room.